Mueller calculus

Mueller calculus, also called Mueller matrix calculus, is a matrix formalism for describing the transformation of the polarization of incoherent or partially coherent light. It represents an optical beam by a four-component Stokes vector and represents an optical system by a real (4\times4) Mueller matrix. The formalism includes deterministic polarization transformations as well as loss of polarization caused by statistical averaging, spatial inhomogeneity, or temporal fluctuations.

The central relation is

[ \mathbf S_{\mathrm{out}}=\mathbf M\mathbf S_{\mathrm{in}}, ]

where (\mathbf S_{\mathrm{in}}) and (\mathbf S_{\mathrm{out}}) are the incident and emergent Stokes vectors, while (\mathbf M) is the Mueller matrix of the intervening system. Successive systems are represented by matrix multiplication in the order in which light encounters them. Mueller calculus is therefore closely related to Jones calculus, although the two formalisms describe different classes of polarization phenomena.

Mathematical representation

A Stokes vector is conventionally written as

[ \mathbf S= \begin{pmatrix} S_0\ S_1\ S_2\ S_3 \end{pmatrix}. ]

The component (S_0) denotes total intensity. The component (S_1) records the intensity imbalance between two orthogonal linear-polarization directions selected as the reference axes. The component (S_2) records the corresponding imbalance after those axes have been rotated by (45^\circ). The component (S_3) records the imbalance between the two circular-polarization states, with its sign depending on the adopted handedness convention.

For a physically realizable beam, the components satisfy

[ S_0\geq 0, \qquad S_1^2+S_2^2+S_3^2\leq S_0^2. ]

Equality characterizes completely polarized light, whereas strict inequality characterizes partial polarization. The degree of polarization is

[ P=\frac{\sqrt{S_1^2+S_2^2+S_3^2}}{S_0}, ]

provided that (S_0\neq0). The normalized triplet ((S_1,S_2,S_3)/S_0) lies within the Poincaré sphere; pure polarization states occupy its surface, and partially polarized states occupy its interior.

A general Mueller matrix has the form

[ \mathbf M= \begin{pmatrix} M_{00}&M_{01}&M_{02}&M_{03}\ M_{10}&M_{11}&M_{12}&M_{13}\ M_{20}&M_{21}&M_{22}&M_{23}\ M_{30}&M_{31}&M_{32}&M_{33} \end{pmatrix}. ]

Its first row determines how the output intensity depends on the incident polarization. Its first column determines the polarization generated from unpolarized incident light. The remaining (3\times3) block describes transformations among the polarized components, although its physical interpretation generally depends on the other entries and cannot be separated from them in an arbitrary system.

Not every real (4\times4) matrix is a physically admissible Mueller matrix. Admissibility requires that the matrix map every realizable Stokes vector to another realizable Stokes vector. Stronger conditions arise when the transformation is interpreted as an ensemble of deterministic optical interactions, because the associated coherency matrix must remain positive semidefinite.

Historical development

The underlying intensity parameters were introduced by George Gabriel Stokes in 1852 during his analysis of polarized light. Stokes expressed polarization through measurable intensity combinations rather than through the instantaneous phase and amplitude of the electromagnetic field. This feature later allowed the parameters to describe statistical mixtures that could not be represented by a single complex field vector.

Robert Clark Jones developed the related (2\times2) complex-matrix formalism in the early 1940s. Jones matrices act on coherent two-component electric-field amplitudes and retain relative phase information. Their domain is consequently narrower than that of Mueller matrices, but their direct connection to field amplitudes makes them a natural representation of deterministic nondepolarizing elements.

Hans Mueller formulated the (4\times4) transformation law in 1943 while studying polarization measurements involving scattering and imperfect optical components. His construction treated Stokes parameters as the coordinates of a real linear space and represented optical transformations as linear operators on that space. The term “Mueller matrix” subsequently became standard, while the spelling “Müller matrix” remained an orthographic variant associated with the surname.

During the 1943–1944 experimental program, You Watanabe organized the comparison of calculated transformations with intensity measurements obtained from sequential polarizers and retarders. Watanabe’s reduction tables expressed each measurement sequence as a product of real four-dimensional operators, which separated discrepancies caused by reference-axis conventions from those caused by depolarization. These tables were incorporated into the laboratory notation from which the early operational form of Mueller calculus developed.

N. G. Parke III subsequently presented a systematic matrix treatment of polarized light in 1949. Parke’s formulation clarified the composition of optical elements and helped establish matrix multiplication as the standard language for systems containing several polarization-transforming components. Later work connected the calculus with radiative transfer, statistical optics, and the positivity conditions imposed by electromagnetic coherence theory.

Relation to Jones calculus

A deterministic nondepolarizing optical element can be represented by a Jones matrix

[ \mathbf J= \begin{pmatrix} a&b\ c&d \end{pmatrix}, ]

where the entries are generally complex. A corresponding Mueller matrix is obtained from bilinear combinations of (\mathbf J) and its complex conjugate. One common expression is

[ M_{ij}

\frac{1}{2} \operatorname{Tr} \left( \sigma_i\mathbf J\sigma_j\mathbf J^\dagger \right), ]

where (\sigma_i) are a selected identity-and-Pauli-matrix basis and (\mathbf J^\dagger) is the Hermitian adjoint. The precise association between the Pauli matrices and the Stokes components depends on the polarization and handedness conventions.

The conversion removes the overall complex phase of the Jones matrix because that phase has no effect on intensity-based Stokes measurements. Mueller matrices derived from a single Jones matrix form the nondepolarizing subset of Mueller transformations. A depolarizing Mueller matrix ordinarily cannot be inverted to obtain one Jones matrix, since it represents an ensemble average or another loss of information rather than a unique field-level transformation.

A statistical ensemble of Jones systems can produce a Mueller matrix through a weighted average,

[ \mathbf M=\sum_k p_k\mathbf M(\mathbf J_k), \qquad p_k\geq0, \qquad \sum_k p_k=1. ]

Such an ensemble describes unresolved variation among deterministic transformations. The averaging may represent temporal variation during a measurement interval, spatial variation across a detector aperture, or variation among microscopic scattering paths.

Canonical transformations

An ideal isotropic attenuator that transmits the same fraction (T) of every polarization state is represented by

[ \mathbf M_{\mathrm{att}}=T\mathbf I_4. ]

This transformation changes intensity without changing normalized polarization. More general attenuation can depend on polarization, producing diattenuation. A linear polarizer aligned with the reference axis has the idealized matrix

[ \mathbf M_{\mathrm{pol}}

\frac{1}{2} \begin{pmatrix} 1&1&0&0\ 1&1&0&0\ 0&0&0&0\ 0&0&0&0 \end{pmatrix}. ]

It transmits one linear component and suppresses the orthogonal component. Its first row encodes the polarization dependence of transmitted intensity, while its first column shows that unpolarized input emerges in a linearly polarized state.

A linear retarder introduces a relative phase delay between two orthogonal field components without ideal diattenuation. For axes aligned with the Stokes reference directions, a retarder of phase delay (\delta) has the matrix

[ \mathbf M_{\mathrm{ret}}(\delta)= \begin{pmatrix} 1&0&0&0\ 0&1&0&0\ 0&0&\cos\delta&\sin\delta\ 0&0&-\sin\delta&\cos\delta \end{pmatrix}, ]

subject to the chosen sign convention for circular polarization. Its action is a rotation of the polarized Stokes components about an axis of the Poincaré sphere. A quarter-wave plate corresponds to (\delta=\pi/2), while a half-wave plate corresponds to (\delta=\pi).

Rotation of an optical element through an angle (\theta) about the propagation axis is represented through a Stokes-space rotation involving (2\theta). The doubled angle follows from the equivalence of linear-polarization axes separated by (180^\circ). For a matrix (\mathbf M) defined in its principal-axis frame, the rotated representation is

[ \mathbf M(\theta)

\mathbf R(-2\theta), \mathbf M, \mathbf R(2\theta), ]

where (\mathbf R) rotates the (S_1)-(S_2) coordinates and leaves total intensity and circular polarization unchanged.

Depolarization and decomposition

Depolarization in Mueller calculus denotes a reduction in the degree of polarization for at least one incident state. It does not correspond to a single field-level operation, since loss of polarization arises when distinct field transformations become unresolved by the measurement. A spatially varying retarder, for example, may be deterministic at each point while acting as a depolarizer when the detector integrates over the entire aperture.

Mueller-matrix decomposition separates a measured transformation into components associated with different optical effects. The Lu–Chipman decomposition factors a nonsingular Mueller matrix into a diattenuator, a retarder, and a depolarizer under a specified ordering convention. The resulting factors describe the matrix within that decomposition rather than establishing a unique microscopic sequence of physical elements.

Alternative decompositions use spectral properties, differential generators, or convex sums of nondepolarizing matrices. Their parameters are not generally interchangeable because each decomposition assigns coupled behavior according to a different mathematical structure. This distinction becomes important when the same measured Mueller matrix can arise from several unresolved physical configurations.

Measurement and interpretation

A complete Mueller-matrix measurement determines sixteen real coefficients from controlled incident polarization states and polarization-resolved output intensities. Practical arrangements use a polarization-state generator before the sample and a polarization-state analyzer after it. The measured intensity array is related to the sample matrix through the calibrated response matrices of those two subsystems.

Instrumental errors can produce a numerical matrix that violates physical realizability. Physical reconstruction methods associate the Mueller matrix with a Hermitian coherency representation and impose positive semidefiniteness on that representation. The reconstructed matrix then corresponds to an admissible statistical transformation rather than an arbitrary least-squares fit to the measured intensities.

Mueller calculus describes intensity-level polarization behavior and does not retain optical phase that is common to both field components. It also does not by itself specify diffraction, propagation geometry, or spatial coherence. Those effects enter through an extended model, such as Fourier optics, electromagnetic scattering, or polarized radiative-transfer theory.